Question

the volume of a rectangular prism can be found by multiplying the base area, \\(b\\), times the height.

if the volume of the prism is represented by \\(15x^2 + x + 2\\) and the height is \\(x^2\\), which expression represents \\(b\\), the area of the base?

options:

• \\(15x + \frac{1}{x} + \frac{x^2}{2}\\)
• \\(15 + \frac{1}{x} + \frac{2}{x^2}\\)
• \\(15x + \frac{1}{x} + \frac{2}{x^2}\\)
• \\(15 + \frac{1}{x} + \frac{x^2}{2}\\)

Explanation:

Identify the given values and formula

We are given the volume of a rectangular prism \(V = 15x^2 + x + 2\) and its height \(h = x^2\). The volume of a rectangular prism is calculated using the formula:

$$V = B \cdot h$$

where \(B\) represents the area of the base.

Set up the equation for the base area

To find the expression representing the base area \(B\), we rearrange the volume formula to solve for \(B\):

$$B = \frac{V}{h}$$

Substituting the given expressions into this formula yields:

$$B = \frac{15x^2 + x + 2}{x^2}$$

Perform the division

Using the Polynomial Division by Monomial knowledge point

$$
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$$

Match with the given options

We compare our simplified expression \(15 + \frac{1}{x} + \frac{2}{x^2}\) with the provided multiple-choice options:

• Option 1: \(15x + \frac{1}{x} + \frac{x^2}{2}\)
• Option 2: \(15 + \frac{1}{x} + \frac{2}{x^2}\)
• Option 3: \(15x + \frac{1}{x} + \frac{2}{x^2}\)
• Option 4: \(15 + \frac{1}{x} + \frac{x^2}{2}\)

The correct expression is represented by the second option.

Answer:

• (A) \(15x + \frac{1}{x} + \frac{x^2}{2}\)
• (B) \(15 + \frac{1}{x} + \frac{2}{x^2}\) (Correct answer)
• (C) \(15x + \frac{1}{x} + \frac{2}{x^2}\)
• (D) \(15 + \frac{1}{x} + \frac{x^2}{2}\)